**Colloquium Publications**

Volume: 54;
2005;
578 pp;
Softcover

MSC: Primary 42; 05; 34;
Secondary 47; 30

Print ISBN: 978-0-8218-4864-7

Product Code: COLL/54.2.S

List Price: $104.00

Individual Member Price: $83.20

**Electronic ISBN: 978-1-4704-3200-3
Product Code: COLL/54.2.E**

List Price: $104.00

Individual Member Price: $83.20

#### This item is also available as part of a set:

#### Supplemental Materials

# Orthogonal Polynomials on the Unit Circle: Part 2: Spectral Theory

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*Barry Simon*

This two-part volume gives a comprehensive overview of the theory of
probability measures on the unit circle, viewed especially in terms of the
orthogonal polynomials defined by those measures. A major theme involves the
connections between the Verblunsky coefficients (the coefficients of the
recurrence equation for the orthogonal polynomials) and the measures, an analog
of the spectral theory of one-dimensional Schrödinger operators.

Among the topics discussed along the way are the asymptotics of Toeplitz
determinants (Szegő's theorems), limit theorems for the density of the
zeros of orthogonal polynomials, matrix representations for multiplication by
\(z\) (CMV matrices), periodic Verblunsky coefficients from the point of view of
meromorphic functions on hyperelliptic surfaces, and connections between the
theories of orthogonal polynomials on the unit circle and on the real line.

The book is suitable for graduate students and researchers interested in
analysis.

#### Readership

Graduate students and research mathematicians interested in analysis.

#### Reviews & Endorsements

This completes with a reader's guide (topics and formulae) and a list of conjectures and open questions. Detailed historic and bibliographic notes are appended to each chapter. A reader is furnished with an extensive notation list and an exhaustive bibliography. The book will be of interest to a wide range of mathematicians.

-- Zentralblatt MATH

The author has now opened new vistas with a monumental treatise that integrates old and new aspects of the subject and describes important connections with mathematical physics, especially with spectral theory of Schrödinger Operations. ... The author has made every effort to guide the reader toward a clear understanding of the material and its various interconnections. The result is an instant classic that will become a standard source for novices and veterans alike.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Orthogonal Polynomials on the Unit Circle: Part 2: Spectral Theory

- Cover Cover11
- Title page iii4
- Dedication v6
- Contents vii8
- Preface to Part 2 xi12
- Notation xiii14
- Chapter 9. Rakhmanov’s theorem and related issues 46724
- Chapter 10. Techniques of spectral analysis 545102
- Chapter 11. Periodic Verblunsky coefficients 709266
- Chapter 12. Spectral analysis of specific classes of Verblunsky coefficients 817374
- Chapter 13. The connection to Jacobi matrices 871428
- Appendix A. Reader’s guide: Topics and formulae 945502
- Appendix B. Perspectives 971528
- Appendix C. Twelve great papers 975532
- Appendix D. Conjectures and open questions 981538
- Bibliography 983540
- Author index 1031588
- Subject index 1039596
- Back Cover Back Cover1604